Unit 2 : Quadratic Functions

Math 2Fall 2017

The Real Number SystemUse properties of rational and irrational numbers.

NC.M2.N-RN.3

Use the properties of rational and irrational numbers to explain why: the sum or product of two rational numbers is rational; the sum of a rational number and an irrational number is irrational; the product of a nonzero rational number and an irrational number is irrational.

The Complex Number SystemDefining complex numbers.

NC.M2.N-CN.1 Know there is a complex number i such that 𝑖2 = – 1, and every complex number has the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers.

Seeing Structure in Expressions Interpret the structure of expressions.

NC.M2.A-SSE.1

NC.M2.A-SSE.1a

Interpret expressions that represent a quantity in terms of its context.a. Identify and interpret parts of a quadratic, square root, inverse variation, or right

triangle trigonometric expression, including terms, factors, coefficients, radicands, and exponents.

NC.M2.A-SSE.1b b. Interpret quadratic and square root expressions made of multiple parts as a combination of single entities to give meaning in terms of a context.

Seeing Structure in Expressions Write expressions in equivalent forms to solve problems.

NC.M2.A-SSE.3Write an equivalent form of a quadratic expression by completing the square, where 𝑎 is an integer of a quadratic expression, 𝑎𝑥2 + 𝑏𝑥+ 𝑐, to reveal the maximum or minimum value of the function the expression defines.

Arithmetic with Polynomial ExpressionsPerform arithmetic operations on polynomials.

NC.M2.A-APR.1 Extend the understanding that operations with polynomials are comparable to operations with integers by adding, subtracting, and multiplying polynomials.

Arithmetic with Polynomial ExpressionsUnderstand the relationship between zeros and factors of polynomials.

NC.M2.A-APR.3 Understand the relationships among the factors of a quadratic expression, the solutions of a quadratic equation, and the zeros of a quadratic function.

Creating EquationsCreate equations that describe numbers or relationships.

NC.M2.A-CED.1Create equations and inequalities in one variable that represent quadratic, square root, inverse variation, and right triangle trigonometric relationships and use them to solve problems.

NC.M2.A-CED.2 Create and graph equations in two variables to represent quadratic, square root and inverse variation relationships between quantities.

NC.M2.A-CED.3 Create systems of linear, quadratic, square root, and inverse variation equations to model situations in context.

Reasoning with Equations and InequalitiesUnderstand solving equations as a process of reasoning and explain the reasoning.

NC.M2.A-REI.1 Justify a chosen solution method and each step of the solving process for quadratic, square root and inverse variation equations using mathematical reasoning.

Reasoning with Equations and InequalitiesSolve equations and inequalities in one variable.

NC.M2.A-REI.4

NC.M2.A-REI.4a

Solve for all solutions of quadratic equations in one variable.a. Understand that the quadratic formula is the generalization of solving 𝑎𝑥2 + 𝑏𝑥+ 𝑐

by using the process of completing the square.

NC.M2.A-REI.4b b. Explain when quadratic equations will have non-real solutions and express complex solutions as 𝑎 ± 𝑏𝑖 for real numbers 𝑎 and 𝑏.

Reasoning with Equations and InequalitiesSolve systems of equations.

NC.M2.A-REI.7 Use tables, graphs, and algebraic methods to approximate or find exact solutions of systems of linear and quadratic equations, and interpret the solutions in terms of a context.

Interpreting FunctionsInterpret functions that arise in applications in terms of the context.

NC.M2.F-IF.4Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: domain and range, rate of change, symmetries, and end behavior.

Interpreting FunctionsAnalyze functions using different representations.

NC.M2.F-IF.7

Analyze quadratic, square root, and inverse variation functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; maximums and minimums; symmetries; and end behavior.

NC.M2.F-IF.8

NC.M2.F-IF.8a

Use equivalent expressions to reveal and explain different properties of a function by developing and using the process of completing the square to identify the zeros, extreme values, and symmetry in graphs and tables representing quadratic functions, and interpret these in terms of a context.

NC.M2.F-IF.9Compare key features of two functions (linear, quadratic, square root, or inverse variation functions) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

Building FunctionsBuild a function that models a relationship between two quantities.

NC.M2.F-BF.1

Write a function that describes a relationship between two quantities by building quadratic functions with real solution(s) and inverse variation functions given a graph, a description of a relationship, or ordered pairs (include reading these from a table).

Building FunctionsBuild new functions from existing functions.

NC.M2.F-BF.3

Understand the effects of the graphical and tabular representations of a linear, quadratic, square root, and inverse variation function f with 𝑘∙𝑓(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥+ 𝑘) for specific values of 𝑘 (both positive and negative).

Math 2 – Transformation Review – Guided Notes

What are the 7 types of basic graphs? (State the name and draw a graph)

Vertical Translation Notes

Example 1 Example 2

Horizontal Translation Notes

Example 1 Example 2

F(x) = - | x + 4 | + 3

Your Turn to graph!

Combining Vertical & Horizontal Notes

Example 1 Example 2

Reflecting Over the X- Axis Notes Reflecting Over the Y- Axis Notes

Class work

1. The graph of y = f(x) is included on each grid below. Use its graph to graph the following functions and describe the transformations:

a. F ( x )= f ( x )−2 b. G ( x )=f ( x+1 ) c. P ( x )=−f ( x )

_________________________ _________________________ _____________________

Video Clip # 2 – Stretching & Shrinking website

d. Q ( x )=1

2f ( x )

e. G ( x )=f −1( x ) f. h ( x )= f (2 x )

________________________ _________________________ __________________________

_____________________ _____________________ ___________________

2. The graph of the parent function is translated two units left and five units up.

Function Parent Function Transformed function g(x)Linear

Quadratic

Absolute value

Rational

Exponentialf(x) =2

x

3. Clearly explain each step required to transform y= f ( x ) into each of the following new functions. Sequence is the same as the operation order.

1. g( x )=f ( x+6 )+7 2. h( x )=−3 f (x−5)−1

3. p( x )=1 /2 f (−x )−4 4. k (x )=−2 f (−x )+10

W # 1 – Parts A – B & Super Hero Character in Google Drive

Part A. The graph of y = f(x) is included on each grid below. Use its graph to graph the following functions:

d. Q ( x )=1

2f ( x )

e. g ( x )=f (−x ) f. h ( x )=2 f ( x ) ______________________________ _________________________ ________________________

g. F ( x )=f ( x )−3 h. G ( x )=f −1( x )

______________________________ ________________________________

Part B. The graph of y = f(x) is included on each grid below. Use its graph to graph the following functions:

a. F ( x )= f ( x )+3 b. G ( x )=f ( x−2 ) c. P ( x )=− f ( x )

d. F ( x )=f ( x )−3 e. g ( x )=f (−x ) f. P ( x )=|f ( x )|

Class workPart C: Write the transformed function: g(x)

The graph of the parent function is vertically stretched by a factor of 2 and is translated left 8 units and down 3 units.

Function Parent Function Transformed function g(x)Square root

Quadratic

Absolute value

Rational

Exponentialf(x) =2

x

Part D Explain each step required to transform from parent function into the new functions.

5: y=1

3( x+6)2+7

6. y=−√x+5+2 Parent function: Parent function:

7. y = −7|4x ─ 3| +9 8. y=−2( x−1)3+4 Parent function: Parent function:

9. y=−2x−4−6 10. y= 1

x−5−3

Parent function: Parent function:

Graph

1. y = 3H(x) 2. y = –2H(x) 3. y = 12H(x)

Points - Old New Points - Old New Points - Old New

Describe Describe Describe

Domain Domain Domain

Range Range Range

This is the function B(x).

4. List its characteristic points.

5. Are these the only points on the graph of B(x)? Explain.

6. What is the domain of B(x)?

7. What is the range of B(x)?

Matching Activity

HW # 2 # 1 - 9

For each of the following, list the effect on the graph of B(x) and then graph the new function.

1. y = B(– x) 2. y = – B(x) 3. y = 13 B(x)

_________________________ _________________________ ___________________________

Domain__________________ ____________________ _____________________

Range___________________ ____________________ _____________________

4. y = 3 B(x) 5. y = B(x – 3) 6. y = B(x + 2) – 1

_________________________ _________________________ ___________________________

Domain__________________ ____________________ _____________________

Range___________________ ____________________ _____________________

7.List the transformations needed to graph the following. Remember that translations are done last.

a. y = 2 F ( x )+2

_________________________________________________________________________________

b. y = 13

F (x−6)

_________________________________________________________________________________

c. y = −F ( x )−12

________________________________________________________________________________

d. y = 3F(-x)

_________________________________________________________________________________

e. y = −¿5F(x)

_________________________________________________________________________________

8. Write an equation for a quadratic function that is shifted right 6 and up 4

9. Write an equation for a quadratic function that is reflected over the x-axis, a vertical stretch of 2, and shifted up 6

Class work – Quadratic Functions

Video Clip Notes – Domain and Range

Example 1: Identify the transformations y= (x−3 )2+5

a. Graph the parent and the new function

Parent Function Domain _________________ Parent Function Range __________________________

New Function Domain __________________ New Function Range ____________________________

Standard Form for a Parabola in Vertex Form:

f(x) – k = A(x – h)2 or y – k = A(x – h)2

Vertex (h, k) “k” can be placed on either side of the equation

Video Clip Notes – Writing the Equation of a Parabola (Video Clip # 9 is optional)

Example 2: Identify the transformations y=−2 ( x )2+4

a. Graph the parent and the new function

Parent Function Domain _________________ Parent Function Range __________________________

New Function Domain ___________________ New Function Range ____________________________

Vertex _______________ Vertex ________________ Vertex ______________

Domain ______________ Domain _______________ Domain _____________

Range _______________ Range ________________ Range ______________HW # 3 – pages 20 - 21

HW # 41. Perform the following operations on the functions.

a. b.b.

c. d.e.

f. g. h.

Video Clip Notes Website HW # 4

HW # 5 (pages 23 – 27)

INVESTIGATION: ADDITION AND SUBTRACTION (pg. 327)My role for this investigation _________________________

If the ticket price is set at x dollars, income and expenses might be estimated as follows. 1. Before each show, the manager uses the function t(x) and s(x) to estimate totalincome.

a. Why does it make sense that each source of income-ticket sales and snack bar sales-might depend on the price set for tickets?

__________________________________________________________________________________________b. What income should the manager expect from tickets sales alone if the ticket price is set at $12? What income from snack bar sales? What income from the two sources combined?

__________________________________________________________________________________________

c. What rule would define the function I(x) that shows how combined income from ticket sales and snack bar sales depends on ticket price? Write a rule for I(x) that is in simplest form for calculation.

__________________________________________________________________________________________

d. How does the degree (the highest exponent of the function) of I(x) compare to the degrees of t(x) and s(x)?

__________________________________________________________________________________________

2. The manager also uses the functions c(x) and b(x) to estimate total operating expenses.

a. Why does it make sense that each source of expense-concert operation and snack bar operation-might depend on the price set for tickets?

__________________________________________________________________________________________

b. What expense should the manager expect from concert operations alone if the ticket price is set at $12? What expense from snack bar operations? What expense from the two sources combined?

__________________________________________________________________________________________

c. What rule would define the function E(x) that shows how combined expense from concert and snack bar operations depends on ticket price? Write a rule for E(x) that is in simplest form for calculation.

__________________________________________________________________________________________

d. How does the degree (the highest exponent of the function) of E(x) compare to the degrees of c(x) and b(x)?

__________________________________________________________________________________________

3. Consider the next function P(x) defined as

a. What does P(x) tell about business prospects for the music venue?

__________________________________________________________________________________________

b. Write two equivalent rules for P(x).

_____________________________________________________________________________________

_____________________________________________________________________________________

c. How does the degree (the highest exponent of the function) of P(x) compare to that of I(x) and of E(x)?

__________________________________________________________________________________________

d. Compare What caution does this result suggest in using subtraction to find the difference of quantities represented by polynomial functions?

__________________________________________________________________________________________

4. a. Test your ideas about the degree of the sum or difference of polynomials by finding the simplest rules for the sum and difference of each pair of functions given below.

i. f(x) = 3p2 - 2p + 3 and g(x) = p2 - 7p + 7

Sum:__________________________________ Difference: ____________________________________

ii. f(x) = 7x2 - 8 and g(x) = 3x2 + 1

Sum:__________________________________ Difference: ____________________________________

iii.

Sum:__________________________________ Difference: ____________________________________

iv.

Sum:__________________________________ Difference: ____________________________________

v.

Sum:__________________________________ Difference: ____________________________________

vi.

Sum:__________________________________ Difference: ____________________________________

b. Explain in your own words how to find the simplest rule for the sum or difference of two polynomials.

__________________________________________________________________________________________

INVESTIGATION: POLYNOMIALS IN CONTEXT

1. Farmer Bob is planting a garden this spring. He wants to plant squash, pumpkins, corn, beans, and potatoes. His plan for the field layout in feet is shown in the figure below. Use the figure and your knowledge of polynomials, perimeter, and area to solve the following:

a. Write an expression that represents the length of the south side of the field.

b. Simplify the polynomial expression that represents the south side of the field.

c. Write a polynomial expression that represents the perimeter of the pumpkin field.

d. Simplify the polynomial expression that represents the perimeter of the pumpkin field. State one reason why the perimeter would be useful to Farmer Bob.

e. Write a polynomial expression that represents the area of the potato field.

f. Simplify the polynomial expression that represents the area of the potato field. State one reason why the calculated area would be useful to Farmer Bob.

2. The Galaxy Sport and Outdoor Gear company has a climbing wall in the middle of its store. Before the store opened for business, the owners did some market research and concluded that the daily number of climbing wall customers would be related to the price per climb x by the linear function n ( x )=100−4 x .

a. According to this function, how many daily climbing wall customers will there be if the price per climb is $10? What if the price per climb is $15? What if the climb is offered to customers at no cost?_______________________________________________________________________________

b. What do the numbers 100 and -4 in the rule of n ( x ) tell about the relationship between climb price and number of customers?

_______________________________________________________________________________

c. What is a reasonable domain for n(x ) tell about the relationship between climb price and number of customers?

Domain: __________________

d. What is the range of n(x ) for the domain you specified in Part c? That is, what are the possible values of n ( x ) corresponding to plausible inputs for the function?

Range: ___________________

e. If the function I (x) tells how daily income from the climbing wall depends on price per climb, why is I ( x )=100 x−4 x2 a suitable rule for that function?

_______________________________________________________________________________

3. The function e (x )=2 x+150 shows how daily operating expenses for the Galaxy Sport climbing wall in the previous problem depend on the price per climb x.

a. Write two algebraic rules for the function P(x ) that gives daily profit from the climbing wall as a function of price per climb, (1) one that shows how income and operating expense functions are used in the calculation of profit and (2) another that is in simpler equivalent form.

P(x )= ______________________ P(x )= ______________________

b. Find P(5). Explain what this result tells about climbing wall profit prospects.

P(5) = _________

c. What is a reasonable domain for P(x ) in this problem situation? Domain: ________________

d. What is the range of P(x ) for the domain you specified in Part c? Range: ________________

e. Write and solve an inequality that will find the climb price(s) for which Galaxy Sport and Outdoor Gear will not lose money on operation of the climbing wall.

Inequality: ___________________ Solution: ______________________

f. Find the price(s) that will yield maximum daily profit from the climbing wall.

_______________________________________________________________________________

Class work Think about the situation!

Let’s say a ball is thrown up into the air. It’s path (or trajectory) would look something like this.

This point represents when the ball reaches the ground. The height on the ground is ______________, so substitute this in for _________. This point is an x-intercept so I can find a solution by _______________ or ______________________.

This point represents when the ball ______________ _________________. At this point, ________ seconds have passed.

This point represents when the ball reaches its ____________________.Key word: _________________, so find the __________________. The x-value represents _____________ and the y-value represents ____________.

Graphing Calculator Activity

Class work – Quadratic Applications1. The product of 2 consecutive integers is 30. Find the integers.

2. Find three consecutive positive odd integers such that the product of the first and the third is 4 less than 7 times the second.

3. An object is moving in a straight line. It initially travels at a speed of 6 m/s, and it speeds up at a constant acceleration of 4 m/s each second. The distance, d, in meters, that this object travels is given by the equation d=2t2 + 6t, where t is in seconds. According to this equation, how long will it take the object to travel 108 meters?

4. In the two squares shown below, the longer square has a side length 1 foot greater than that of the smaller square. If the combined area of the two squares is 113 ft2, find the length of the side of the smaller square.

5. An object is launched into the air from a ledge 16 feet off the ground at an initial velocity of 96 feet per second. Its height H, in feet, at t seconds is given by the equation H=-16t2 + 96t + 16. Find all times t that the object is at a height of 160 feet off the ground.

6. Jason jumped off of a cliff into the ocean in Acapulco while vacationing with some friends. His height as a function of time could be modeled by the function, h ( t )=−16 t2+16 t+480, where t is the time in seconds and h is the height above water, in feet.

a) How long did it take for Jason to reach his maximum height?

b) What was the highest point that Jason reached?

c) How long did it take for Jason to hit the water?

7. The square of a positive number increased by 4 times the number is equal to 140. Find the number.

HW # 6 (Part 1 & 2)

Part 11. A rocket is launched from atop a cliff. It’s height as a function of time is modeled by h (t )=−16 t 2+116 t+101. How long will it take for the rocket to hit the ground after takeoff?

2. You and a friend are hiking in the mountains. You want to climb to a ledge that is above you. The height of the grappling hook you throw is given by h ( t )=−16 t 2−32 t+5. What is the maximum height of the hook? The ledge is 20 feet above you. Will you be able to throw the hook onto the ledge?

3. You are trying to dunk a basketball, and need to jump 2.5 ft in the air to dunk that ball. The height that your feet are above ground is given by the function h ( t )=−16 t 2+12t . What is the maximum height your feet will be above ground? Will you be able to dunk the ball?

Part 2: How much the graph of y = x2 be changed to produce each of the following graphs? Write shift up, shift down, shift left, shift right, narrower or flatter in the blank. If more than one change is needed, you may write up to 3 of these options in the blank.

1. y=x2+5 1. _____________________________________

2. y=( x+2)22. _____________________________________

3. y=( x−9 )23. _____________________________________

4. y=−4 x24. _____________________________________

5. y=−( x−4 )2+2 5._____________________________________

6.y= 1

3x2

6. _____________________________________

HW 8 & On – line Lesson Review Clip

1. Simplify: −4 y2 (5 y4−3 y2+2 )2. Multiply:

3. Compute the product: 1. Multiply to remove the parenthesis:

2. Simplify: 3. Simplify:

Multiply the following polynomials.

7. 8.

9. 10.

11. Simplify 12. What is the product of and

13. The expression is equivalent to

(1) (3)

(2) (4)

14. The expression is equivalent to

(1) (3)

(2) (4)

HW 7 Parts 1 – 2 Graded HW # 2 Name ___________________________________

Part 1: How much the graph of y = x2 be changed to produce each of the following graphs? Write shift up, shift down, shift left, shift right, narrower or flatter in the blank. If more than one change is needed, you may write up to 3 of these options in the blank.

7. y=( x+2)2−3 7. _____________________________________

8. y=2(x−4 )28. _____________________________________

9.y=−1

2x2+1

9. _____________________________________

10. y=x2−3 10. ____________________________________

11.y= 2

5( x+2)2+1

11. ____________________________________

12. y=3 ( x−1)2−2 12. ____________________________________

Part 24. A diver is standing on a platform. He jumps and his height is shown by h ( t )=−16 t2+8 t+24. How long will it take for him to hit the water after jumping?

5. A ball is thrown upward. It’s height is modeled by h (t )=−16 t 2+5t +15. How long will it take for the ball to hit the ground?

6. A ship drops anchor in a harbor. Its function is h ( t )=−16 t 2+49, where h is the height above the water surface and t is seconds.a) How high above the surface of the water is the anchor when it is dropped?

b) How many seconds will it take the anchor to hit the water?

Solving Quadratics by Factoring & Graphing

1. Solve the following parabola equations algebraically (factor & solve).

a. -x2+8=−17 Graphically Table

b. x2−16=33 Graphically Table

c. x2+11=47 Graphically Table

X Y00

0

X Y00

0

X Y00

0

2. Fill in the chart and graph the following parabolas: y = polynomial.

Polynomial y – interceptFactored

FormZerosRoots

Minimum

x2 - 4x – 5

Polynomialy –

interceptFactored

FormZerosRoots

Minimum

x2 + 2x – 8 = 0

Polynomialy –

interceptFactored

FormZerosRoots

Maximum

-x2 + 2x + 15 = 0

HW 9 # 1 - 15

Factor the GCF FIRST, then factor the polynomial, if possible.

1. Factor completely: 3 x2+15 x−42 2. Express as the product of two binomial factors.

3. What is the greatest common factor of

and ?4. Factor completely:

5. Factor completely: 6. Factor completely:

7. Factor completely: 8. Factor Completely:

9. What is the greatest common factor of

and ?10. Factor completely:

11. What are the factors of ? 12. Written in simplest factored form, the

binomial can be expressed as(1) 2(x – 5)(x – 5) (3) (x – 5)(x + 5)(2) 2(x – 5)(x + 5) (4) 2x(x – 50)

13. Factor completely: 14. If is one factor of , what is the other factor?

15. Use the diagram below to answer the following questions:a. Find the area of the shaded region, in terms of x.

b. If the area of the shaded region is 40 , what is the value of x?

Class work

Sketch the graphs of the following functions, clearly marking all intercepts, the axis of symmetry, and the vertex of the curve.

1. y = (x - 2)(x + 4)

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

2. y = (x-1)(x-6)

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

APPLICATIONS WITH PARABOLIC FUNCTIONS

Using the graph at the right, It shows the height h

in feet of a small rocket t seconds after it is launched.

The path of the rocket is given by the equation:

h = -16t2

+ 128t.

1. How long is the rocket in the air? _________

2. What is the greatest height the rocket reaches? ____

3. About how high is the rocket after 1 second? _______

4. After 2 seconds,a. about how high is the rocket?_________

b. is the rocket going up or going down? ________

5. After 6 seconds,a. about how high is the rocket? _______

b. is the rocket going up or going down? ________

6. Do you think the rocket is traveling faster from 0 to 1 second or from 3 to 4 seconds? Explain your answer.

7. Using the equation, find the exact value of the height of the rocket at 2 seconds.

h (height (feet))

time (seconds)8765432

250

200

150

100

1

50

HW 10 # 1 - 3

1. Match up each of the graphs below with the following functions:

(a) y = x2 - 2 (d) y = (x - 2)(x + 1) (b) y - 2 = (x + 1)2 (e) y = - x2 + 2 (c) y = (x + 2)(x - 1) (f) y = x2 + 2

(i) (ii) (iii)

(iv)

(v) (vi)

Sketch the graphs of the following functions, clearly marking all intercepts, the axis of symmetry, and the vertex of the curve.

2. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

3. A ball is thrown in the air. The path of the ball is represented by the equation

h = -t2

+ 8t. Graph the equation over the interval 0 t 8 on the accompanying grid.

What is the maximum height of the ball?_______________

How long is the ball above 7 meter? ________________

Class work – Vertex Form

Quadratic Equation:

a

height (meters)

time (seconds)

h

k

Vertex

1. Find the vertex, then find two additional values to graph!

a) b) c)

Vertex: _______ Vertex: _______ Vertex: _______

Working backwards: You know the transformation, you have to write the equation.

d) is stretched vertically by 3 and translated left 2

e) is reflected across the x-axis and

translated 3 units up

Hint : Find the x – value for the axis of symmetry !

a. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

b. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

a. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

b. y =

b. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

HW 11(a, c, & Take notes on online HW 11)

a. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

b y =

b. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

Class work

c. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

d. y =

d. y =

a. x – int ______________________________

b. y – int ______________________________

c. Axis of Symmetry ____________________

d. Vertex _____________________________

e. Domain _____________________________

f. Range ______________________________

HW 12 # 1 - 10

7. A record label uses the function to model the sales of a new release. The number of albums sold is a function of time, t, in days. On which day were the most albums sold? What is the maximum number of albums sold on that day?

8.

You learned how to find the zeros of a quadratic function. Today we are going to look at a real world application of this:

___Projectile______ motion

#9: A soccer ball is kicked from the ground level with an initial velocity of 32 ft/s. After how many seconds will the ball

hit the ground?

Step 1: Write the general projectile function

Step 2: Plug in anything you can

Step 3: When does it mean for the ball to hit the ground? Answer: ____________________________!

#10: Marilyn hit a golf ball on the ground with her driver. Use the general function for a projectile to write a function

that shows the height in feet of her golf ball as a function of time. The ball was hit with an initial vertical velocity of 100

feet per second.

How long will Marilyn’s ball stay in the air?

Quiz Review # 1 - 10

Directions: Solve by factoring:

is 0 (disappears) when…How hard you kick/throw/shoot the object from the beginning

1) 2)

3. Use completing the square to put this equation in vertex form. Then find the vertex.

Vertex form: ___________________________ Vertex:_________________

For the problems below, make sure your answers are in the right format (point, line, number, etc.)!Find the information below then sketch the graph:

4.

which way does it open?_______________

Has a MAX or MIN?_______________

Vertex: _________________

Axis of Symmetry:_______________

Max / Min value:________________

Use the info to sketch the graph:

5.

which way does it open?_______________

Has a MAX or MIN?_______________

Vertex: _________________

Axis of Symmetry:_______________

Max / Min value:________________

Use the info to sketch the graph:

6. Here’s a graph…tell me the info:

Which way does it open?

Has a MAX or MIN?_________

Vertex: _________

Axis of Symmetry:________

Max / Min value:

y-intercept:____________

x-intercepts:__________

Given that a = -1, write the equation in vertex form:__________________________

7. Factor each of the following.

a) b)

_________________________ _____________________________

c) d)

__________________________ _______________________________

e) f)

_____________________________ _____________________________

8. For each problem below, you are given the roots of an equation. Find the factors, then write the equation in standard form.

a) 3 and – 4 b) -17 and –5

________________________ ____________________________

9. Use your calculator to find the following information for the quadratic function .

Vertex is________________ zeros are _________________

10. The late great Mr. Rogers dropped a ball in the Land of Make Believe off the top of the castle that has a height of 50ft.

a) Write a function that describes the position of the ball as a function of time.

b) Determine when the ball hits the ground.

Using the Discriminant Quadratic equations can have two, one, or no solutions (x-intercepts). You can determine how many solutions a

quadratic equation has before you solve it by using the ________________.

The discriminant is the expression under the radical in the quadratic formula: Discriminant = b2 – 4ac

If b2 – 4ac < 0, then the equation has 2 imaginary solutions

If b2 – 4ac = 0, then the equation has 1 real solution

If b2 – 4ac > 0, then the equation has 2 real solutions

A. Finding the number of x-intercepts Determine whether the graphs intersect the x-axis in zero, one, or two points.

1.) 2.)

B. Finding the number and type of solutionsFind the discriminant of the quadratic equation and give the number and type of solutions of the equation.

3.) 4.)

5.) 9x2 – 6x = 1 6.) 4x2 = 5x + 3

Draw a small example of each of the following.1. b2-4ac < 0, then there are ________ roots and the graph looks like _____________

2. b2-4ac > 0, then there are ________ roots and the graph looks like _____________

3. b2-4ac = 0, then there are ________ roots and the graph looks like _____________

HW 13Nature of the Roots Worksheet

1. Use factoring, quadratic formula, and a graphing calculator to graph and find the roots.

Function

y= x2+5x+50 y=x2+4x+4 Y=x2+5x-50

Graph

Roots

D = b2 – 4ac(ax2 + bx + c)

D = D = D =

Function

y=x2+4x+1 y=x2+4x+100 y=4x2+4x+1

Graph

Roots

D = b2 – 4ac(ax2 + bx + c)

D = D = D =

Function

y=x2+3x+6 y=x2+4 y=x2-4

Graph

Roots

D = b2 – 4ac(ax2 + bx + c)

D = D = D =

Class Work & Homework 14 = Quadratic Formula and Complex Numbers

A. Review of Simplifying Radicals and Fractions

Simplify expression under the radical sign ( ); reduce Reduce only from ALL terms of the fraction. (You can’t reduce a number outside of a radical with a number inside of a radical) Make sure that you have TWO answers

Simplify:

1.) 2.)

3.) 4.)

5.) 6.)

B. Solving Quadratics using the Quadratic Formula

Solve by using the quadratic formula:

7.) x2 + x = 12

(std. form):

a = _____

b = _____

c = _____

b2 – 4ac = _______

8.) 5x2 – 8x = -3

(std. form):

a = _____

b = _____

c = _____

b2 – 4ac = _______

9.) -x2 + x = -1 10.) 3x2 = 7 – 2x

11.) -x2 + 4x = 5 12.)

Solving using the quadratic formula: Put into standard form (ax2 + bx + c = 0) List a = , b = , c =

Plug a, b, and c into

Simplify all roots (look for ); reduce

Solve using the SQUARE ROOT PRINCIPAL.

( x+3 )2=5 Use the square root principal.

√ ( x+3 )2=±√5 Simplify.

x+3=±√5 Solve for x by subtracting 3.

x=−3±√5

Examples

13. x2=49 14. x

2=20 15. x2+14=50

Solve each equation.

16. 17. 18.

19. x2−5=3 20. ( x+4 )2=8 21. 8 x2=200

THE MORAL OF THE STORY: Try the factoring method or solving by the square root principal first. If all else fails… use the Quadratic Formula.

#22: A rocket is launched from the ground with an initial velocity of 144 ft/sec. (All work is to be done without using a calculator to graph. You may check your answers by graphing.)

a. Write a function that represents this situation.

b. At what time(s) will the rocket be 320 feet in the air?

c. What is the maximum height of the rocket? How long does it take for the rocket to reach this height?

d. When will the rocket hit the ground?

Simplify the following

23.

10+√505 24.

−2+√124 25.

8+√186 26.

11+√12111

I. Solve by Factoring27.) x2 – 64 = 0 28.) x2 – 6x – 16 = 0 29.) x2 + 3x = 40

30.) 2 x2+3 x+1=0 31.) x² + 6x = 0

II. Solve by Square Roots

32.) x2=64 33.) 4 x2=81 34.) x

2+7=−300 35.) (x – 5)2 = 36

III. Solve by using the quadratic formula:36. 4x² – 8x = 1 37. x² + 8x = 0

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

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-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

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-2

2

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-10 -8 -6 -4 -2 2 4 6 8 10

-10

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-4

-2

2

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-10 -8 -6 -4 -2 2 4 6 8 10

-10

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-2

2

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-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

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-2

2

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-10 -8 -6 -4 -2 2 4 6 8 10

-10

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-2

2

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-10 -8 -6 -4 -2 2 4 6 8 10

-10

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-10 -8 -6 -4 -2 2 4 6 8 10

-10

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-2

2

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10

Class workMatch each given function with the graph on the right-hand side.

1. ______ y=2 x+1

2. ______ y=x2−x−6

3. ______ y=−x2+x+6

4. ______ y=2 x−1

5. ______ y=x2+ x−6

6. ______ y=−2 x−1

7. ______ y=−x2−5 x−6

8. ______ y=−2x+1

DC

BA

HG

FE

SOLVING QUADRATIC-LINEAR SYSTEMSTwo equations will be given to you with the directions to solve the system graphically.

One equation will be a quadratic. This equation has degree ________ The second equation will be linear. This equation has degree ________

Where the two graphs ___________________, this is your ______________________.

There are three possible situations as answers illustrated below. Indicate the number of solutions in each representation.

Examples:1. Solve the following system of equations graphically and check.

y=−x2+4 x−3x+ y=1

y=x2+4 x+4y=−2x+4

3. The graphs of the equations y=x2 and x=2 intersect in:

(1) 1 point (2) 2 points (3) 3 points (4) 4 points

4. Which is a solution or the following system of equations?y=2 x−15y=x2−6 x

(1) (3, –9) (2) (0, 0) (3) (5, 5) (4) (6, 0)

5. When the graphs of the equations y=x2−5 x+6 and x+ y=6 are drawn on the same set of axes, at which point do the graphs intersect?

(1) (4, 2) (2) (5, 1) (3) (3, 3) (4) (2, 4)

HW 15

SOLVING QUADACTIC –LINEAR SYSTEMS ALGEBRAICALLY

Steps:

1. Make sure both equation are in y = form if necessary

2. Substitute the linear equation into the ‘y part’ of the quadratic equation, to

have only one variable left to solve in the equation.

3. Get NEW quadratic equation into standard form (_________________________)

and______________________________

4. Since it is a quadratic: Must FACTOR TO SOLVE FOR X.

(How many answers should you get?_______)

5. Must find other variable (y) by substituting your ‘x’ answers into one of the equation and solve for y.

6. Check solutions

Examples:

1. Solve the following system:

y=x2−x+2y=2 x

2. Find the solutions of:

y=−x2+4 x−3x+ y=1

3. Solve for the solutions:

y=x2−7 x+13x− y=2

Class workSample Problem 1:

Graph the following system of equations: y = 2x –3 and y = -x + 3.

Solution:

STEP #1. Graph both equations on the same coordinate plane.

Input for screen of graphing calculator.

View of Graph Table of Values

When x=2, the values of y are the same. This is the solution.

Extra step to find the intersection:Identify the location of the point or points where the two lines intersect.

Problem 2:Graph following system of equations:

Use the same steps as problem no. 1:Draw s rough sketch below: Make a table of values from the Graphing Calculator Table:

Extra STEP to find the intersection:Identify the location of the points where the two graphs intersect.

Practice Problems

1.

Solve by Hand:

On the set of axes below, solve the following system of equations graphically and state the coordinates of all points in the solution set.

2. Solve the following systems of equations graphically, on the set of axes below, and state the coordinates of the point(s) in the solution set.

HW 16 – # 1 - 3

1. On the set of axes below, solve the following system of equations graphically for all values of x and y.

To check on your graphing calculator (find intersection):

Go to (Calculate) and pick (intersection)

Move cursor to wanted intersection point Enter Enter Enter

2nd Trace #5

2. The following graph shows how income and operating cost depend on ticket price and how they are related to each other.

a. Use the graph to estimate answers for the following questions, and explain how you arrive at each estimate. i. For what ticket price(s) will operating cost exceed income? _______________________

ii. For what ticket price(s) will income exceed operating cost? _______________________

iii. For what ticket price(s) will income equal operating cost? _______________________

b. Write an equation that helps in locating the break-even point(s)-the ticket prices for which income exactly equals operating cost.

_____________________________________________________________________________

3. It is likely that the show producers want to do more than break-even. They will probably seek maximum profit.a. Use the income and operating cost functions to write a function showing how profit depends on ticket price. Write the function in two equivalent forms – one that shows the expressions for income and cost and the other that is the simples for calculating the profit. ____________________________________________________

b. Use the profit function to estimate the maximum profit plan – the ticket prices that will lead to maximum profit and the dollar profit that will be made at that price.___________________________________________

c. Use the results from Part b to calculate the number of tickets sold and the operating cost in the maximum profit situation. ____________________________________________________________________________

SYSTEMS OF LINEAR INEQUALITIES

Up to this point, you know how to graph simple inequalities like

Today we’re going to graph systems of inequalities.

Step 1: If necessary, put each inequality in slope-intercept form. Reminder: If you multiply or divide by a negative number you have to change the sign!

Step 2: Graph each inequality. If the inequality sign is ≥ or ≤ use a solid line to connect your points.

If the inequality sign is > or < use a dashed line to connect your points.Label your point(s) of intersection (if there are any)

Step 3: Shade the solution set for each individual inequality.

When you graph an inequality, Draw the line, Solid or Dashed, Pick a point, and Shade what’s true, That is all you dooo.

The solution to the system of inequalities is the doubly shaded region.

Graph the solution set to the system of inequalities on the axes below to check your answer.

x

y

Graph the solution set to the system of inequalities on the axes below to check your answer.

x

y

HW 17 pages 67 - 691. Solve each of the following equations algebraically.

c. Sketch graphs of the linear and quadratic functions involved in part 1 above and explain how the graphs illustrate the solutions.

i. ii. iii.

SYSTEMS OF LINEAR-QUADRATIC INEQUALITIES

You solve these in exactly the same way that you solved linear inequalities. The only difference is that there will be one line and one parabola in on your graphs!

x

y

(4)

x

y

(5)

7.) What are the two mistakes in setting up the quadratic formula:

Solve: 2x² – x – 6 = 0x=

−1±√(−1 )2−4(2 )(6)2(2)

Solve the equation correctly below:

Look at my Super Hero

For the Project you may hand draw or use the computer to complete the project.Prezi, Power point, Youtube video, comic strip, Word Document, Poster, Story Book, etc, All work must be shown to get full credit.Groups 1 person = 2 flights 2 people = 3 flights and questions adjusted

Project Check list & Grading ScaleEach project can earn 0, ½, or 1 point in each of the following categories:

1 – Super Hero, Name, Brief by – line story, Nemisis, Name, Brief by – line story.2 – Colorful characters, background, and well labeled graph.3 – Questions 1 – 3 answered with work shown.4 – Questions 4 – 5 answered with work shown.5 – Questions 6 – 7 answered with work shown.6 – Question 8 answered with work shown.7 – Questions 9 – 10 answered with work shown.8 – Neat and On time with student(s) names and class period labels.9 – Format used 10 – Judges responsesExtra Point – The top best 3 presentations voted on by the teacher and the judges.

Graph the problem and label appropriately.

1) How high did (Your Super Hero) reach? ___________2) How high did (Your Super Hero) start from the ground? _____3) At approximately what time did (Your Super Hero) reach a height of feet? ____4) What is the parent function of the graph above?5) Represent the parent function through a T – Table and a Graph.

6) 6) At approximately what time did (Your Super Hero) hit the ground? _____7) At seconds, what was the height of (Your Super Hero)? ____

8) What is the equation of (Your Super Hero’s) flight path?

9) Your arch nemesis fires an arrow that follow a different path that passes through your start from ground level. What is the equation of this path?

10) Will your nemesis catch (Your Super Hero)? Explain and show your work to your answer in relationship to (Your Super Hero).

Story Line and Work Space for Brainstorming:

Review

Solve each of these equations, sketch graphs showing the functions involved, and label points corresponding to solutions with their coordinates.

1. 2.

3

.

4.

5. Solve the following system algebraically: 6. Graphically solve the system of equations in the problem to the left.

7. Which graph could be used to find the solution to the following system of equations?

(1) (2) (3) (4)

REVIEW Cont.:

8. 8. The graph of is shifted up 3 units and right 5 units. Which equation represents the resulting graph?

9. Which graph displays the function ?

a. c.

b. d.

a. c.

b. d.

10. The height of a swimmer’s dive off a 10-foot platform into a diving pool is modeled by the equation

, where x represents the number of seconds since the swimmer left the diving board and y represents the number of feet above or below the water’s surface. What is the farthest depth below the water’s surface that the swimmer will reach? a. 6 feet c. 10 feet

b. 8 feet d. 12 feet

Solve each equation any way you want. Show your work.

16. x2 + 11x + 18 = 0 19. x2 + 2x + 1 = 15 22. 7x2 – 9x +1 = 0

17. (x + 2)² = 36 20. x² – 10x + 25 = 0 23. x² + 3x + 7 = 0

18. x² = 36 21. x² – 6x + 2 = 0 24. x² – 5x + 4 = 0